The part I don't understand is why the surface integral over the surface ΔS now calculated for the field at time t - shouldn't it be B(t+Δt)? In the lecture (at 1:02:15), he says it doesn't matter because ΔS is first order in time, it's proportional to Δt. And the difference* is going to be proportional to Δt so we don't have to worry about the change in time over an infinitesimal surface because it's second order in time.

I'm not sure why the Δs in the first integral become derivatives (shouldn't we then have dt rather than Δt in front of the integral?), but I think these are all meant to be infinitesimals.
And another thing which relates to my first question is why B in the second integral is no longer time-dependent.

Still not sure I quite understand it yet...I'm not sure if I've seen the product rule derived that way before. I'm unfamiliar with the idea of 'infinitesimals of higher order'.

I just noticed an error in my original post, the difference between the integrals should be ∬S B(t+Δt).dA−∬S B(t).dA, so over the original surface. So the total change in flux is given by

∬S B(t+Δt).dA−∬S B(t).dA + ∬ΔS B(t).dA

and of course B(t) in the last integral is what I can't quite grasp. It gives the flux through the change in surface, so if we want to calculate the total change, why not calculate it at the later time?
Are we saying that the magnetic field cannot 'that fast'? That is, in the time it takes for the surface to change by ΔS, the magnetic field cannot have changed a lot from what it was at B(t)?
Seems a bit dodgy to me to be honest :p

Are there any textbooks that discuss this (at this level)? I've yet to come across one that does so without using full vector calculus (and on this note, which books presents Faraday's law in the form given in the lecture?)

Are we saying that the magnetic field cannot change 'that fast'? That is, in the time it takes for the surface to change by ΔS, the magnetic field at B(t+Δt) cannot have changed a lot from what it was at B(t)?
Seems a bit dodgy to me to be honest :p